Show that the linearization of f(x) = (1 + x)ᵏ at x = 0 is L(x) = 1 + kx.

Verified step by step guidance

1. To find the linearization of a function at a point, we use the formula L(x) = f(a) + f'(a)(x - a), where a is the point of linearization. Here, a = 0.
2. First, evaluate f(x) at x = 0. For f(x) = (1 + x)ᵏ, substitute x = 0 to get f(0) = (1 + 0)ᵏ = 1.
3. Next, find the derivative of f(x) = (1 + x)ᵏ with respect to x. Using the power rule, f'(x) = k(1 + x)^(k-1).
4. Evaluate the derivative at x = 0. Substitute x = 0 into f'(x) to get f'(0) = k(1 + 0)^(k-1) = k.
5. Substitute f(0) and f'(0) into the linearization formula: L(x) = 1 + k(x - 0) = 1 + kx. This shows that the linearization of f(x) at x = 0 is L(x) = 1 + kx.

Key Concepts

Linearization

Linearization is the process of approximating a function near a specific point using its tangent line. For a function f(x), the linearization at a point a is given by L(x) = f(a) + f'(a)(x - a). This method is particularly useful for simplifying complex functions into linear forms that are easier to analyze and compute.

Derivative

The derivative of a function measures the rate at which the function's value changes as its input changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In the context of linearization, the derivative at a point provides the slope of the tangent line, which is essential for constructing the linear approximation.

Binomial Theorem

The Binomial Theorem provides a formula for expanding expressions of the form (a + b)ⁿ, where n is a non-negative integer. It states that (a + b)ⁿ = Σ (n choose k) a^(n-k) b^k, where the sum is taken over k from 0 to n. In the given function f(x) = (1 + x)ᵏ, this theorem helps in understanding the behavior of the function around x = 0, particularly in determining its value and derivative at that point.